Application of Fourier Bessel transform
and time-frequency based method for
extracting rotating and maneuvering
targets in clutter environment
T. Thayaparan
DRDC Ottaw a
P. Suresh
Sri Sathya Sai University
Defence R&D Canada – Ottawa
Technical Memorandum
DRDC Ottawa TM 2013-153
August 2014
Application of Fourier Bessel transform and
time-frequency based method for extracting
rotating and maneuvring targets in clutter
environment
T. Thayaparan
Defence Research and Development Canada – Ottawa
P. Suresh
Sri Sathya Sai University
Defence Research and Development Canada – Ottawa
Technical Memorandum
DRDC Ottawa TM 2013-153
August 2014
c Her Majesty the Queen in Right of Canada (Department of National Defence), 2014
°
c Sa Majesté la Reine en droit du Canada (Ministère de la Défense nationale), 2014
°
Abstract
In this paper, we report the efficiency of Fourier Bessel transform and time-frequency
based method in conjunction with the fractional Fourier transform, for extracting
micro-Doppler radar signatures from the rotating targets. This approach comprises
mainly two processes; the first being decomposition of the radar return in order to extract micro-Doppler (m-D) features and the second being the time-frequency analysis
to estimate motion parameters of the target. In order to extract m-D features from
the radar signal returns, the time domain radar signal is decomposed into stationary
and non-stationary components using Fourier Bessel transform in conjunction with
the fractional Fourier transform. The components are then reconstructed by applying
the inverse Fourier Bessel transform. After the extraction of the m-D features from
the target’s original radar return, time-frequency analysis is used to estimate the target’s motion parameters. This proposed method is also an effective tool for detecting
manoeuvring air targets in strong sea-clutter and is also applied to both simulated
data and real world experimental data. Results demonstrate the effectiveness of the
proposed method in extracting m-D radar signatures of rotating targets. Its potential
as a tool for detecting, enhancing low observable manoeuvring and accelerating air
targets in littoral environments is demonstrated.
Résumé
Le présent rapport décrit l’efficacité de la méthode fondée sur l’analyse temps-fréquence
et la transformée de Fourier-Bessel, de concert avec la transformée de Fourier fractionnaire, pour extraire les signatures radar obtenues par microdécalage Doppler dans
les cibles rotatives. Cette approche comprend principalement deux processus, le premier étant la décomposition des échos radar pour extraire les caractéristiques du
microdécalage Doppler, et la seconde étant l’analyse temps-fréquence pour évaluer
les paramètres de déplacement des cibles. Afin d’extraire les caractéristiques du microdécalage Doppler dans les échos radar, les signaux radar du domaine temporel
sont divisés en éléments fixes et non fixes à l’aide de la transformée de FourierBessel, de concert avec la transformée de Fourier fractionnaire. Les éléments sont
alors reconstitués en utilisant la transformée inverse de Fourier-Bessel. Une fois les
caractéristiques extraites de l’écho radar original des cibles, l’analyse temps-fréquence
permet d’évaluer les paramètres de déplacement des cibles. Cette méthode proposée
constitue également un outil efficace de détection des cibles aériennes manIJuvrables
dans un important fouillis de mer. Il est également utilisé pour les données simulées
et les données expérimentales réalistes. Les résultats démontrent l’efficacité de cette
méthode pour extraire les signatures radar obtenues par microdécalage Doppler des
cibles rotatives. Son potentiel comme outil de détection et de poursuite de cibles
aériennes manIJuvrables et furtives dans des milieux littoraux est démontré.
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Executive summary
Application of Fourier Bessel transform and
time-frequency based method for extracting rotating
and maneuvring targets in clutter environment
T. Thayaparan, P. Suresh; DRDC Ottawa TM 2013-153; Defence Research and
Development Canada – Ottawa; August 2014.
Background: Today radar technology has attained a broad scope of applications
ranging from military to civilian. Target classification is one such area, which investigates both the moving characteristics as well as discrimination of targets. Recent
research indicates that the detection of an unknown deterministic signal in a high
noise environment is of crucial interest in many real-world applications. In the case
of a stationary signa,l a sinusoidal signal with constant frequency, for example, the
Fourier transform (FT) method concentrates all the signal energy in one frequency
point while the noise is uniformly distributed over all frequencies. Thus, it is easy
to conclude that the FT-based detection method provides the optimal detection in
the case of stationary signal. However, for non-stationary signals, i.e., when the frequency content of a signal changes over time, the spectral content of such signals
becomes time-varying, and thus the FT-based detector will not provide the optimal
result. The time-frequency formulation of the FT, that is, by using a window in
the time domain, the short time Fourier transform (STFT) has the same advantages
and drawbacks similar to FT. Therefore, there is a need for more sophisticated timefrequency tools for the analysis of highly non-stationary signals. In this report, we
present a high-resolution analysis approach for extracting rotating and maneuvring
targets in heavy clutter environment.
Results: We present the efficiency of Fourier Bessel transform and time-frequency
based method in conjunction with the fractional Fourier transform, for extracting
micro-Doppler radar signatures from the rotating targets. This approach comprises
mainly two processes; the first being decomposition of the radar return, in order to
extract micro-Doppler (m-D) features and the second being, the time-frequency analysis to estimate motion parameters of the target. In order to extract m-D features
from the radar signal returns, the time domain radar signal is decomposed into stationary and non-stationary components using Fourier Bessel transform in conjunction
with the fractional Fourier transform. The components are then reconstructed by applying the inverse Fourier Bessel transform. After the extraction of the m-D features
from the target’s original radar return, time-frequency analysis is used to estimate
the target’s motion parameters. This proposed method is also an effective tool for
DRDC Ottawa TM 2013-153
iii
detecting manoeuvring air targets in strong sea-clutter and is also applied to both
simulated data and real world experimental data. Results demonstrate the effectiveness of the proposed method in extracting m-D radar signatures of the rotating
targets. Its potential as a tool for detecting, enhancing low observable manoeuvring
and accelerating air targets in littoral environments is demonstrated.
Significance: Micro-Doppler features have great potential for use in automatic target classification algorithms. Although there have been studies of m-D effects in
radar in the past few years, the proposed approach has great potential for use in
target identification applications. As such, this report contributes additional experimental m-D data and analysis, which should help in developing a better picture of the
m-D research and its applications to indoor and outdoor radar detection and automatic gait recognition systems. The method developed in this study can also be used
to evaluate the motion parameters of the rotating antenna on a ship or ground using
RADARSAT data. Alternatively, this approach can also be used to extract biometric information related to periodic contraction of a heart, blood vessels, lungs, other
fluctuations of the skin in the process of breathing and heart beating, which should
help in human m-D research and its applications to through-wall radar imaging.
The results from high-frequency surface-wave radar (HFSWR) data clearly show that
the proposed approach outperforms the traditional Fourier-based and time-frequency
methods in terms of good detection and false alarm rates for non-stationary signals.
The method presented here is not restricted to this particular application, but it can
also be applied in various other settings of non-stationary signal analysis and filtering. More generally, it is believed that the time-frequency formulation of optimum
detection can provide new hints for handling open problems in a comprehensive way.
iv
DRDC Ottawa TM 2013-153
Sommaire
Application of Fourier Bessel transform and
time-frequency based method for extracting rotating
and maneuvring targets in clutter environment
T. Thayaparan, P. Suresh ; DRDC Ottawa TM 2013-153 ; Recherche et
développement pour la défense Canada – Ottawa ; août 2014.
Contexte : La technologie radar d’aujourd’hui donne lieu à une grande diversité
d’applications militaires et civiles. La classification des cibles est l’une de ces applications. Elle porte sur l’examen des caractéristiques de déplacement et la discrimination des cibles. De récentes recherches révèlent que la détection de signaux
déterministes inconnus dans un milieu très bruyant est cruciale dans de nombreuses
applications concrètes. Dans le cas d’un signal fixe, un signal sinusoïdal avec une fréquence constante (p. ex., transformée de Fourier) concentre toute l’énergie du signal
en un point de fréquence alors que le bruit est réparti uniformément sur l’ensemble
des fréquences. Ainsi, il est facile de conclure que la méthode fondée sur la transformée de Fourier offre la meilleure détection avec ce type de signaux. Toutefois,
pour les signaux non fixes (p. ex., lorsque le contenu fréquentiel d’un signal change
au fil du temps), le détecteur fondé sur la transformée de Fourier n’offrira pas les
meilleurs résultats puisque le contenu spectral de tels signaux varie avec le temps.
La formulation temps-fréquence de la transformée de Fourier (c’est à-dire l’utilisation
d’une fenêtre pour le domaine temporel, la transformée de Fourier à temps court)
présente des avantages et des désavantages semblables à la transformée de Fourier.
Ainsi, des outils complexes de temps-fréquence sont nécessaires pour l’analyse des signaux extrêmement non fixes. Dans le présent rapport, nous présentons une approche
analytique à haute résolution pour extraire des cibles rotatives et manIJuvrables dans
un important fouillis.
Résultats principaux : Le présent rapport décrit l’efficacité de la méthode fondée
sur l’analyse temps-fréquence et la transformée de Fourier-Bessel, de concert avec la
transformée de Fourier fractionnaire, afin d’extraire des signatures radar obtenues
par microdécalage Doppler des cibles rotatives. Cette approche comprend principalement deux processus, le premier étant la décomposition des échos radar pour extraire
les caractéristiques du microdécalage Doppler, et la seconde étant l’analyse tempsfréquence pour évaluer les paramètres de déplacement des cibles. Afin d’extraire les
caractéristiques du microdécalage Doppler des échos radar, les signaux radar du domaine temporel sont divisés en éléments fixes et non fixes à l’aide de la transformée
de Fourier-Bessel, de concert avec la transformée de Fourier fractionnaire. Les éléments sont alors reconstitués en utilisant la transformée inverse de Fourier-Bessel.
DRDC Ottawa TM 2013-153
v
Une fois les caractéristiques extraites de l’écho radar original des cibles, l’analyse
temps-fréquence permet d’évaluer les paramètres de déplacement des cibles. Cette
méthode proposée constitue également un outil efficace de détection des cibles aériennes manoeuvrables dans un important fouillis de mer. Il est également utilisé
pour les données simulées et les données expérimentales réalistes. Les résultats démontrent l’efficacité de cette méthode pour extraire les signatures radar obtenues par
microdécalage Doppler des cibles rotatives. Son potentiel comme outil de détection et
de poursuite de cibles aériennes manIJuvrables et furtives dans des milieux littoraux
est démontré.
Portée des résultats : Les caractéristiques du microdécalage Doppler ont un grand
potentiel dans les algorithmes de classification automatique des cibles. Bien qu’il y
ait eu des études sur les effets du microdécalage Doppler dans le domaine du radar au
cours des dernières années, l’approche proposée présente un grand potentiel dans les
applications d’identification des cibles. En ce sens, le présent rapport fait part de nouvelles données expérimentales et d’une analyse du microdécalage Doppler qui devrait
aider à l’obtention d’un meilleur tableau de la recherche sur le microdécalage Doppler
et de ses applications dans les systèmes radar de détection et les systèmes automatiques de reconnaissance du mouvement, intérieurs et extérieurs. La méthode abordée
dans la présente étude servira à évaluer les paramètres de déplacement de l’antenne
rotative installée sur un navire ou au sol à partir des données de RADARSAT. Elle
servira aussi à extraire les renseignements biométriques propres à la contraction périodique du cIJur, des vaisseaux sanguins et des poumons et aux mouvements de la peau
durant la respiration et les battements du cIJur. Toutes ces données devraient aider à
la recherche sur le microdécalage Doppler à l’égard de l’humain et à ses applications
dans le domaine de l’imagerie radar passe-muraille.
Les données du radar haute fréquence à ondes de surface (RHFOS) démontrent clairement que l’approche proposée surpasse les méthodes classiques de temps-fréquence
et de Fourier en ce qui concerne la bonne détection et les taux de fausses alarmes
pour des signaux non fixes. La méthode présentée ici ne se limite pas à cette application particulière. Elle peut également être appliquée dans divers autres contextes
d’analyse et de filtrage des signaux non fixes. En général, on croit que la formulation temps-fréquence d’une détection optimale peut fournir de nouveaux indices pour
gérer des problèmes ouverts de façon exhaustive.
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Table of contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 Time-Frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1
Linear Time-Frequency Transforms . . . . . . . . . . . . . . . . . . .
3
2.1.1
Short-Time Fourier Transform . . . . . . . . . . . . . . . . .
3
Quadratic Time-Frequency Transforms . . . . . . . . . . . . . . . .
3
2.2.1
Wigner-Ville distribution . . . . . . . . . . . . . . . . . . . .
4
3 Fourier-Bessel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4 Fractional Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . .
5
5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
6.1
Rotating reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
6.2
Rotating antenna in SAR . . . . . . . . . . . . . . . . . . . . . . . .
15
6.3
Manoeuvring air target in sea-clutter . . . . . . . . . . . . . . . . .
18
6.3.1
Filtering in Frequency domain . . . . . . . . . . . . . . . . .
20
6.3.2
Filtering using FB-TF method . . . . . . . . . . . . . . . . .
22
7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
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List of figures
Figure 1:
a) STFT of the multi component signal, and b)WVD of the multi
component signal. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Figure 2:
FB Coefficients of the multi component signal. . . . . . . . . . . .
8
Figure 3:
(a-c) are WVD of first, second and third LFM chirp, respectively.
d) FB-WVD plot of multi component signal. . . . . . . . . . . . .
9
Separation of two LFM components using Fractional Fourier
Transform and Fourier-Bessel Transform. . . . . . . . . . . . . . .
9
Figure 5:
Picture of the target simulator experimental apparatus. . . . . . .
11
Figure 6:
a) TF signature of the signal from one rotating corner reflector
facing the radar, b) TF signature of the extracted oscillating
signal, and c) TF signature of the extracted body signal. . . . . .
12
a) TF signature of the signal from two rotating corner reflector
facing the radar, b) TF signature of the extracted oscillating
signal, and c) TF signature of the extracted body signal. . . . . .
13
a) TF signature of the signal from three rotating corner reflector
facing the radar, b) TF signature of the extracted oscillating
signal, and c) TF signature of the extracted body signal. . . . . .
14
Top- the original SAR image at range cell; bottom- zoomed in
SAR image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Figure 10: The Fourier Transform of the original time series. . . . . . . . . .
16
Figure 11: a) TF signature of the original signal, b) TF signature of the
extracted oscillating signal, and c) TF signature of the extracted
body signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Figure 12: Path of the King-Air 200 as a function of range (in km) and
azimuth (in degrees). . . . . . . . . . . . . . . . . . . . . . . . . .
18
Figure 13: a) FT of the signal 1: non-accelerating target far from Bragg’s
lines, b) FT of the signal 2: accelerating target far from Bragg’s
lines, and c) FT of the signal 3: target very close to Bragg’s lines.
19
Figure 14: Band-rejection filter. . . . . . . . . . . . . . . . . . . . . . . . . .
20
Figure 4:
Figure 7:
Figure 8:
Figure 9:
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DRDC Ottawa TM 2013-153
Figure 15: a) STFT of signal 1, b) STFT of signal 1 after sea clutter is
removed, c) STFT of signal 2, d) STFT of signal 2 after sea
clutter is removed, e) STFT of signal 3, f) STFT of signal 3 after
sea clutter is removed, g) STFT of the signal 4, and h) STFT of
the signal 4 after sea clutter is removed. . . . . . . . . . . . . . . .
21
Figure 16: FB coefficients of the signal 1. . . . . . . . . . . . . . . . . . . . .
22
Figure 17: Figures (a,d,g), (b,e,h),(c,f,h) show the results of STFT,
FB-STFT and FB-WVD analysis for three signals, respectively. .
23
Figure 18: a) STFT representation of the original signal, b) FB-STFT
representation of the target, and c) FB-WVD representation of
the target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
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1 Introduction
Radar signals can be analyzed either in the time domain or in the frequency domain.
The Fourier transform (FT) is the most widely used tool for analyzing signals in
frequency domain. The standard FT decomposes a signal into its frequency components and gives the relative strength of each component. Since radar signals are
non-stationary in nature, their spectral content changes over a period of time. For
non-stationary signal analysis, FT is not the prefered choice, as it does not provide
any information about time. Hence, joint time-frequency techniques can be used as a
tool for analyzing non-stationary signals. Joint time-frequency representations transform a one-dimensional time domain signal into a two-dimensional time-frequency
representation, thus enabling easy display and study of time-varying frequencies. An
important advantage of the time-frequency representations is the ease with which
the target signals can be identified. Most widely used time-frequency transforms
are short-time Fourier transform (STFT) and Wigner Ville distribution (WVD). In
STFT, time and frequency resolutions are limited by the size of window function
used in calculating STFT. For mono-component signals, WVD gives the best time
and frequency resolutions without any cross terms. However, in the case of multi
component signals, the occurrence of interference terms degrade the readability of
the time-frequency representation and limits the usefulness of WVD. In WVD, cross
terms arise due to the interference among the auto-terms of the signal.
In order to achieve cross-term free WVD, Pachori et al. in [1], [2] and [3] used
Fourier-Bessel transform to decompose the multi-component signal, and then applied
WVD to each component separately to analyze its time-frequency distribution. This
approach is applied to the multi-component signal whose signal components overlap
only in the time domain. This is applied to the simulated data. But if the components of a multi-component signal overlap in both time and frequency domains then it
is not possible to separate the signal components using the method in [1], [2] and [3].
However, in real-time applications, several scenarios are related to a multi-component
signal whose signal components overlap in both time and frequency domains. Therefore, FBT and WVD alone can not be used in real-time applications. This paper
presents a new approach, which is based on Fourier Bessel transform in conjunction
with the Fractional Fourier transform (FrFT) to decompose the non-stationary signal whose component frequencies overlap in both time and frequency domains. The
WVD is then applied to each component separately to analyze its time-frequency
distribution. This approach is an advancement to the method used by [1] and [2]
and has now several real-time applications. We have successfully demonstrated the
proposed approach with experimental data sets.
In order to extract micro-Doppler (m-D) features from the radar signal returns, the
time domain radar signal has to be decomposed into stationary and non-stationary
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components. This can be achieved by applying FBT and FrFT to the radar returns
and choosing the Fourier-Bessel (FB) coefficients corresponding to the stationary
and the non-stationary components. The stationary and the oscillating signal can
be reconstructed by applying the inverse Fourier-Bessel transform (IFBT) on the
selected FB coefficients. After the separation of the m-D features from the target’s
original radar return, time-frequency analysis is then used to estimate the motion
parameters of the target. We report here the application of Fourier-Bessel and TimeFrequency (FB-TF) based method for the analysis of High-Frequency Surface-Wave
radar (HFSWR) signals. Conventionally, targets are detected from radar signals by
the FT or by Doppler processing method. If the target is constantly accelerating, FT
can still be used to detect the target and estimate its median velocity, provided the
acceleration is small [4]. However, if the target is highly accelerating, the performance
of the Fourier method deteriorates as the spectrum gets smeared. The degree of
smearing increases, when the number of pulses increase for a given acceleration or
when acceleration increases for a given number of pulses [4]. If the smearing is too
high, the Fourier method can even fail to detect the target. The case of highly
accelerating targets correspond to the analysis of signals with fast time variations
of the frequency content. Since time-frequency representations display time-varying
frequencies, this kind of signal should be analyzed by time-frequency representations
rather than FT [5]. Time-Frequency based decomposition provides an extraction of
individual signal components and is also efficient in separating the target signal from
an undesirable clutter [6]. In the case of HFSWR signals, where the sea clutter signal
is very strong compared to the target signal, FBT and FrFT can be used to separate
the target signal from the sea clutter. Time-frequency transforms can be used for the
detection and tracking of low observable maneuvering and accelerated targets in the
littoral environments.
This report is organized into six sections. In section II, a brief introduction to TimeFrequency analysis, particularly STFT and WVD is presented. Sections III and IV
deal with the mathematical formulation of FBT and Fractional Fourier transform
(FrFT). In section V, the application of Fourier-Bessel and Time-Frequency (FB-TF)
method, in removing the interference terms that occur when a multi-component signal
is analyzed using WVD, is presented. Section VI demonstrates the effectiveness of
the proposed method in extracting m-D features of the rotating targets and also in
the reduction of sea clutter, thus enhancing the target detection.
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2 Time-Frequency analysis
Time-Frequency techniques are broadly classified into two categories: Linear transforms and Quadratic (or bilinear) transforms.
2.1 Linear Time-Frequency Transforms
All those time-frequency representations that obey the principle of superposition
can be classified under the linear Time-Frequency transforms. Some of the linear
Time-Frequency transforms are STFT, Continuous Wavelet transform (CWT) and
the Adaptive Time-Frequency transforms. STFT is the most widely used time frequency technique among the linear Time-Frequency transformations.
2.1.1 Short-Time Fourier Transform
The basic principle behind STFT is segmenting the signal into narrow time intervals
using a window function and taking Fourier transform of each segment.
   (  ) =
Z∞
() ( −  ) exp(−2 )
(1)
−∞
Where () is the signal to be analyzed and  (), windowing function centered at
 =  . STFT has limited time-frequency resolution which is determined by the size of
the window used. The uncertainty principle prohibits the usage of arbitrarily small
duration and small bandwidth windows. A fundamental resolution trade-off exists: a
smaller window has a higher time resolution but a lower frequency resolution, whereas
a larger window has a higher frequency resolution but a lower time resolution. Hence,
STFT is not capable of analyzing transient signals that contain high and low frequency
components simultaneously.
2.2 Quadratic Time-Frequency Transforms
Cohen, in 1966, showed that all the existing bilinear time-frequency distributions
could be written in a generalized time-frequency form. In addition, this general form
can be used to facilitate the design of new time-frequency transforms. The definition
of the Cohen’s class distribution function is as [13] follows
 ( ) =
Z∞ Z∞
 (  )(  ) exp(−2( −   ))
(2)
−∞ −∞
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where (  ) is the kernal function and  (  ) is the ambiguity function which is
defined as follows.
Z∞
( +  2)∗ ( −  2)
(3)
 (  ) =
−∞
If (  ) = 1, we obtain WVD. The prominent members of Cohen’s class include
WVD, Pseudo Wigner-Ville distribution, Choi-Williams distribution, cone-shaped
distribution and adaptive kernel representation.
2.2.1 Wigner-Ville distribution
The WVD was originally developed in the area of quantum mechanics by Wigner [11]
and then introduced for signal analysis by Jean Ville [12]. It is defined as:
  ( ) =
Z∞ Z∞
 (  ) exp(−2( −  ))
(4)
−∞ −∞
Compared to STFT, WVD has much better time and frequency resolution. But the
main drawback of the WVD is the cross-term interference. This interference phenomenon shows frequency components that do not exist in reality and considerably
affect the interpretation of the time frequency plane. Cross-terms are oscillatory in
nature and are located midway between the two components [13]. Presence of crossterms severely limits the practical applications of WVD. Various modified versions of
WVD have been developed to reduce cross-terms. These techniques include distributions from Cohen’s class by Cohen (1989), Non-linear filtering of WVD by Arce and
Hasan (2000), S-Method by Stankovic (1994), Polynomial WVD by Boashash and
O’Shea (1994). The application of Fourier-Bessel transform to obtain a cross term
free WVD distribution is explained in section V.
3 Fourier-Bessel Transform
The FBT decomposes a signal in to a weighted sum of an infinite number of Bessel
functions of zeroth- order. Mathematically, the FBT  () of a function () is
represented as [7] :
Z∞
(5)
 () = 2  ()0 (2)
0
() = 2
Z∞
 ()0 (2)
(6)
0
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where 0 (2) are the zeroth-order Bessel functions and  is transform variable.
FBT is also known as Hankel transform. As the FT over an infinite interval is related
to the Fourier series over a finite interval, so the FBT over an infinite interval is
related to the FB series over a finite interval. FB series expansion of a signal (), in
the interval (0  ) is given as [1]:
() =

X
 0 (
=1

) 0    

(7)
FB coefficients,  are computed by using following equation.
 =
R
2 ()0 (  )
0
 2 [1 ( )]2
(8)
where  , r = 1,2,3,...M are the ascending order positive roots of 0 () = 0. Since
Bessel function supports a finite bandwidth around a center frequency, the spectrum
of the signal can be represented better using FB expansion. As the Bessel functions form orthogonal basis and decay over the time, non-stationary signals can be
better represented using FB expansion [8]. It turns out to be a one-to-one relation
between frequency content of the signal and the order of the FB expansion, where
the coefficients attain maximum amplitude [9]. As the center frequency of the signal
is increased, it is observed that the order of the FB Coefficients is increased. Similarly there is a relationship between the bandwidth of the signal and the range of FB
Coefficients. In particular, the range of FB Coefficients increases with the increase
in the bandwidth of the signal[10]. Since both amplitude modulation (AM) and frequency modulation (FM) are part of the Bessels’s basis function, the FB expansion
can represent the reflected signal from a rotating target more efficiently.
4 Fractional Fourier Transform
Fractional Fourier transform (FrFT) is the generalization of the classical Fourier transform. The applications of FrFT can be found in signal processing, communications,
signal restoration, noise removal and in many other science disciplines. It is a powerful tool used for the analysis of time-varying signals. The FrFT is a linear operator
that corresponds to the rotation of the signal through an angle i.e. the representation
of the signal along the axis u, making an angle a with the time axis. The a th order
Fractional Fourier Transform of the function f(u) is defined as [14]:
Z
(9)
a (u) = (u 0 )a (u u 0 )du 0 
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5
a (u u 0 ) =  [i(cot u 2 ) − 2 csc  uu 0 + cot u 2 ]
where
a
=
2
p
 = 1 − i cot 
(10)
(11)
(12)
For a = 1, we find that  = 2 ,  = 1 and
1 () =
Z∞
exp(−20 )(0 )0
(13)
−∞
for a = 0, FrFT reduces into identity operation. For a = 1, FrFT is equal to standard
FT of f(u). For a = -1, FrFT becomes an inverse FT. FrFT can transform a signal
either in time or in frequency domain into a domain between time and frequency.
FrFT depends on the parameter a and can be interpreted as rotation by an angle
a in the time-frequency plane. The FrFT of a signal can also be interpreted as a
decomposition of the signal in terms of chirps [15].
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5 Simulation Results
In this section, we demonstrate the application of the proposed method by removing
the cross terms in the WVD representation of a multi-component signal. Consider a
discrete time domain signal, s[n], which is sum of the three linear chirps given by:
[] =
3
X
=1
1
 exp(2  +   ( )2 )
2
(14)
where  are the amplitudes of the constituent signals,  are the fundamental frequencies,   are chirp rates and T is the sampling interval. Figure 1a and Figure
1b show the STFT and WVD representations of the multi component signal in the
equation 14.
(a)
(b)
Figure 1: a) STFT of the multi component signal, and b)WVD of the multi compon-
ent signal.
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7
Figure 2: FB Coefficients of the multi component signal.
Table 1:
signal Required FB Coefficients
chirp 1
(1-45)
chirp 2
(108-150)
chirp 3
(151-230)
From Figure 1a, it is evident that STFT representation of the signal is free from cross
terms but its time and frequency resolutions are poor. As expected WVD gives good
time and frequency resolution but is corrupted with the occurrence of cross terms. In
order to remove these cross terms, the signal is analyzed using FBT. FB coefficients
are calculated using equation (8). Figure 2 shows the FB coefficients of the multi
component signal. By taking the significant order of the FB coefficients, the multi
component signal can be decomposed into its individual components. Table 1 shows
the order of the significant FB coefficients that are selected for each chirp signal.
Individual components are reconstructed by applying IFBT using the selected FB
coefficients. Figure 3a, 3b and 3c show the WVD representation of each component
of the multi-component signal. Figure 3d shows the plot obtained by adding WVD
representations of the three linearly frequency modulated (LFM) signals together.
Results in Figure 3 show that the occurrence of cross terms in WVD can be eliminated, if the multi component signal is decomposed into its individual components, by
expanding the signal using FB series and applying WVD to the constituent signals
separately. Using FBT, we can separate the components of the multi-component
signals, if their frequencies do not over lap in the frequency domain. But if their
frequencies overlap in time and/or frequency domain, it is not possible to separate
them using FBT. By using FrFT and FBT, we can separate the components of the
multi-component signal whose frequencies overlap in time andor frequency domain.
8
DRDC Ottawa TM 2013-153
(a)
(c)
(b)
(d)
Figure 3: (a-c) are WVD of first, second and third LFM chirp, respectively. d)
FB-WVD plot of multi component signal.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Separation of two LFM components using Fractional Fourier Transform
and Fourier-Bessel Transform.
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9
Figure 4a shows the STFT representation of the two LFM signals whose frequencies
overlap in the frequency domain. Time-frequency characteristics of the signal was
rotated by 36 in the clockwise direction by using FrFT, such that their frequency
components do not overlap in the frequency domain. Figure 4b displays the STFT
representation of the signal after rotation. Now using the FBT, the two frequency
components of the multi-component signal were separated. Figures 4c and 4d show
the separated components of the signal. After the separation of the components, timefrequency characteristics of the signal was rotated by 36 in the counterclockwise
direction using FrFT. Figures 4e and 4f show the separated LFM components. It
should be emphasized here that this approach works well for any number of chirps
with different angles.
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Figure 5: Picture of the target simulator experimental apparatus.
6 Experimental Results
In this section, we demonstrate the application and effectiveness of the FB-TF method
with five different types of radar data obtained in various scenarios.
6.1 Rotating reflectors
Experimental trials were conducted to investigate and determine the m-D radar signatures of targets using an X-band radar. The target used for this experimental trial
was a spinning blade with corner reflectors attached. These corner reflectors were
designed to reflect electromagnetic radiation with minimal loss. These controlled
experiments can simulate the rotating type of objects, generally found in an indoor
environment such as a rotating fan and in an outdoor environment such as a rotating
antenna or rotors. Controlled experiments allow us to set the desired rotation rate of
the target, to cross check and assess the results.
A picture of the target is shown in Figure 5. This experiment was conducted with
a radar operating at 9.2 GHz and the pulse repetition frequency (  ) was 1 kHz.
The target employed in this experiment was at a range of 300 m from the radar and
the distance between the two reflectors was 38 inches. The corner length of the reflector was 10 inches with a side length of 12 inches. STFT representation is utilized
in order to depict the m-D oscillation. Figure 6a shows the STFT representation
of the signal obtained from one rotating corner reflector facing the radar. From the
DRDC Ottawa TM 2013-153
11
(a)
(b)
(c)
Figure 6: a) TF signature of the signal from one rotating corner reflector facing the
radar, b) TF signature of the extracted oscillating signal, and c) TF signature of the
extracted body signal.
12
DRDC Ottawa TM 2013-153
(a)
(b)
(c)
Figure 7: a) TF signature of the signal from two rotating corner reflector facing the
radar, b) TF signature of the extracted oscillating signal, and c) TF signature of the
extracted body signal.
DRDC Ottawa TM 2013-153
13
(a)
(b)
(c)
Figure 8: a) TF signature of the signal from three rotating corner reflector facing the
radar, b) TF signature of the extracted oscillating signal, and c) TF signature of the
extracted body signal.
14
DRDC Ottawa TM 2013-153
time-frequency signature, we can observe that the m-D of the rotating corner reflector
is a time-varying frequency spectrum. Figure 6a clearly shows the sinusoidal motion
of the corner reflector. The second weaker oscillation represents the reflection from
the counter weight that was used to stabilize the corner reflector during the operation. It also contains a constant frequency component which is due to reflection from
stationary body of the corner reflector. FBT was utilized in order to separate stationary component from the rotating component. Figure 6b shows the time-frequency
signature of the extracted oscillating signal. Figure 6c displays the time-frequency
signature of the extracted body signal. The rotation rate of the corner reflector is
directly related to the time interval of the oscillations. From the additional time
information, the rotation rate of the corner reflector is estimated at about 60 rpm.
Similar analysis was done for the signals collected from two and three corner reflectors. Figure 7a shows the STFT representation of the original signal from two corner
reflectors where as Figures 7b and 7c show the time-frequency representations of the
extracted oscillating signal and the extracted body signal respectively. In this case,
the rotation rate of the corner reflector was 40 rpm. Figure 8a displays the STFT
representation of the signal when the target is rotating with three corner reflectors.
Figures 8b and 8c show the time-frequency representations of the extracted oscillating signal and extracted body signal respectively. The estimated rotation rate of the
corner reflector was about 60 rpm. Rotation rates estimated by the time-frequency
analysis agree with the actual values.
6.2 Rotating antenna in SAR
Radar returns were collected from a rotating antenna using a APY-6 radar in a SAR
scenario. Using these data sets, the m-D features relating to a rotating antenna were
extracted. The m-D features for such rotating targets may be seen as a sinusoidal
phase modulation of the SAR azimuth phase history. The phase modulation may
equivalently be seen as a time-varying Doppler frequency [19].
Figure 9 () shows the original SAR image and Figure 9 () displays the
zoomed in SAR image between the range cells 115 and 130. The Doppler smearing
due to the rotating parts is often well localized in a finite number of range cells [19].
It is reasonable to process the Doppler signal for each range cell independently. Since
the prior information about the location of the target is known, the data at the range
cell 123 was analyzed using the FB-TF method. The FT of the original time series
at range cell 123 is shown in Figure 10. The rotating antenna is located close to
the zero Doppler and cannot be detected using FT method. Original time series was
decomposed using FBT and rotating and stationary components of the signal were
captured by different order of FB coefficients. Stationary signal and oscillating signals
were reconstructed by applying IFBT on the selected coefficients.
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15
Figure 9: Top- the original SAR image at range cell; bottom- zoomed in SAR image.
Figure 10: The Fourier Transform of the original time series.
16
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(a)
(b)
(c)
Figure 11: a) TF signature of the original signal, b) TF signature of the extracted
oscillating signal, and c) TF signature of the extracted body signal.
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17
Figure 12: Path of the King-Air 200 as a function of range (in km) and azimuth (in
degrees).
Figure 11a illustrates the time-frequency signature of the original signal, and Figure
11b displays the time-frequency signature of the extracted oscillating signal where as
Figure 11c illustrates the time-frequency signature of the extracted body signal. Using
the time-frequency plot, the rotation rate of the antenna is estimated by measuring
the time interval between the peaks. The period is the time interval between peaks
[19]. As an example in Figure 11b, there are three peaks. The time interval between
peak 1 and 2, between 2 and 3, and between 1 and 3 were measured. The average
value was then used to estimate the rotation rate. The estimated rotation rate is 4.8
seconds, which is very close to the actual value of 4.7 seconds.
6.3 Manoeuvring air target in sea-clutter
The signals used in the following analysis were collected from the experimental air
craft (King- Air 200). It was performing manoeuvres and being tracked by a high
frequency surface wave radar (HFSWR) with a 10 - element linear receiving antenna
array. The HFSWR was operating at 5.672 MHz and scans were performed at a pulse
repetition frequency of 9.17762 Hz. Each trial corresponded to a block of 256 pulses.
Therefore, the coherent integration time (CIT) of each signal was 27.89 sec. As
shown in Figure 12, the King-Air performed two figure-eight manoeuvres. Each one
consisted of two circles with an approximate diameter of 10 km. The first figure-eight
manoeuvre was performed at 200 ft (61m), while the second was performed at 500 ft
(152m). As shown in Figure 12, the location of the King-Air was marked by a square
18
DRDC Ottawa TM 2013-153
(a)
(b)
(c)
Figure 13: a) FT of the signal 1: non-accelerating target far from Bragg’s lines, b)
FT of the signal 2: accelerating target far from Bragg’s lines, and c) FT of the signal
3: target very close to Bragg’s lines.
DRDC Ottawa TM 2013-153
19
Figure 14: Band-rejection filter.
when each signal was collected. Each signal reflected a different scenario that could
arise when tracking a manoeuvring aircraft. Since the sea clutter is stronger than
the target signal, detecting a target in the presence of the sea clutter is a challenging
problem. For efficient detection and extraction of the target features, target signal
has to be separated from the sea clutter and should be analyzed using time-frequency
analysis. One way to separate the target signal from the sea clutter is to use digital
filtering techniques in Frequency domain.
6.3.1 Filtering in Frequency domain
The Fourier spectra of the three signals are shown in Figures 13a, 13b and 13c. We
observe that the target signal is buried in the background consisting of clutter and
noise (thermal and atmospheric). Here the sea clutter is due to Bragg scattering
from the surface of the ocean [18]. The Fourier spectra contained two large spectral
lines around the zero Doppler and sea clutter components were concentrated around
zero doppler. Figure 13c clearly illustrates that when the target is accelerating close
to zero frequency or there is sea clutter, the FT method fails to provide optimum
detection performance [6].
Since the sea clutter appears around zero Doppler, it can be removed using digital
filtering techniques in the frequency domain. Figure 14 shows the band-rejection
filter that was used to filter the sea clutter. Figures 15a, 15c and 15e show the STFT
plots of the three signals respectively. Figures 15b, 15d and 15f show the results of
separating target from the sea clutter using the band-rejection filter.
20
DRDC Ottawa TM 2013-153
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 15: a) STFT of signal 1, b) STFT of signal 1 after sea clutter is removed, c)
STFT of signal 2, d) STFT of signal 2 after sea clutter is removed, e) STFT of signal
3, f) STFT of signal 3 after sea clutter is removed, g) STFT of the signal 4, and h)
STFT of the signal 4 after sea clutter is removed.
DRDC Ottawa TM 2013-153
21
Figure 16: FB coefficients of the signal 1.
Table 2:
Signals Sea Clutter Coefficients Target Coefficients
Signal 1
(1-25)
(120-138)
Signal 2
(1-25)
(44-84)
Signal 3
(1-25)
(141-199)
The above results demonstrate that target signal and sea clutter can be separated
using filtering techniques in the frequency domain, although these filtering techniques
fail to separate the target signal from the sea clutter when the target signal crosses the
sea clutter. Figure 15g shows the STFT representation of the target signal crossing
the sea clutter and Figure 15h shows the STFT representation of the target signal,
after the sea clutter is removed using band-rejection filter. The above results show
that it is not possible to separate the target signal and sea clutter if the target is
crossing the sea clutter. In the next section, a method to separate the target signal
and sea clutter even when the target signal crosses the sea clutter is proposed.
6.3.2 Filtering using FB-TF method
Radar returns were analyzed using FBT and FB coefficients were calculated using
equation 8. Figure 16 shows the plot of the FB Coefficients of the signal 1. We
can observe that returns from the sea clutter were captured by the lower order FB
coefficients and that the target signal was captured by the higher order FB coefficients
of Fourier-Bessel basis functions. Since target signal and sea clutter are captured by
different orders of FB coefficients, we can easily separate the target from the sea
clutter. Table 2 contains selected FB coefficients for sea clutter and target for three
signals.
22
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Figure 17: Figures (a,d,g), (b,e,h),(c,f,h) show the results of STFT, FB-STFT and
FB-WVD analysis for three signals, respectively.
Target signal was reconstructed by applying IFBT on the selected FB coefficients of
the target. After the target signal is separated from the sea clutter, time-frequency
representations like STFT and WVD were used to extract more information from it.
Plots in the Figure 17 shows the results of STFT and FB-STFT methods for signals
1, 2 and 3. By using FBT, we can separate the target signal from the sea clutter
more efficiently even when the target signal is very close to sea clutter. In the case
of target signal crossing the sea clutter, as shown in the Figure 18, it is possible to
separate them using FrFT and FBT. By using FrFT, time-frequency signature of the
signal is rotated in counter clockwise direction through an angle  such that, the
target signal is aligned perpendicular to the frequency axis at around zero doppler.
Now the signal is analyzed using FBT and the target signal is separated by selecting
the higher order FB coefficients corresponding to the target signal. Time-frequency
(TF) signature of the target signal is reconstructed by applying IFBT on the selected
FB coefficients. Now the TF signature of the target signal is rotated in the clockwise
direction through an angle  to obtain the separated target signal. Figure 18b and
Figure 18c shows the FB-STFT and FB-WVD representations of the signal after the
target is separated from sea clutter.
DRDC Ottawa TM 2013-153
23
Figure 18: a) STFT representation of the original signal, b) FB-STFT representation
of the target, and c) FB-WVD representation of the target.
24
DRDC Ottawa TM 2013-153
7 Conclusion
This paper presents a FB-TF based approach for m-D analysis, for the extraction
of m-D features of the radar returned signals from the rotating targets, both in
SAR and ISAR scenario. By applying the proposed method to simulated and several
experimental data sets, the effectiveness of this FB-TF technique was confirmed. This
method combines both FBT and time-frequency analysis to extract the m-D features
of the radar returns. By applying the proposed method to the rotating antenna
data and to the rotating corner reflectors data, the potential of the proposed method
is ascertained. From the extracted m-D signatures, information about the target’s
micro-motion dynamics such as rotation rate is obtained. The experimental results
agree with the expected outcome. FB-TF proves to be a useful tool in the reduction
of the sea clutter and target enhancement. Using FB-TF method, we could separate
the target from the strong sea clutter. In the case of target signal crossing the sea
clutter, target signal was separated from the sea clutter using the FrFT and FBT.
Results demonstrate that the proposed method could be used as a potential tool for
detecting and enhancing low observable maneuvering, accelerating air targets in the
littoral environments.
DRDC Ottawa TM 2013-153
25
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Application of Fourier Bessel transform and time-frequency based method for extracting
rotating and maneuvring targets in clutter environment
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Thayaparan, T.; Suresh, P.
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announcement audience may be selected.)
UNLIMITED
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ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly
desirable that the abstract of classif ed documents be unclassif ed. Each paragraph of the abstract shall begin with an indication of the
security classif cation of the information in the paragraph (unless the document itself is unclassif ed) represented as (S), (C), or (U). It is
not necessary to include here abstracts in both off cial languages unless the text is bilingual.)
In this paper, we report the efficiency of Fourier Bessel transform and time-frequency
based method in conjunction with the fractional Fourier transform, for extracting microDoppler radar signatures from the rotating targets. This approach comprises mainly two
processes; the first being decomposition of the radar return in order to extract microDoppler (m-D) features and the second being the time-frequency analysis to estimate
motion parameters of the target. In order to extract m-D features from the radar signal
returns, the time domain radar signal is decomposed into stationary and non-stationary
components using Fourier Bessel transform in conjunction with the fractional Fourier
transform. The components are then reconstructed by applying the inverse Fourier Bessel
transform. After the extraction of the m-D features from the target’s original radar return,
time-frequency analysis is used to estimate the target’s motion parameters. This proposed
method is also an effective tool for detecting manoeuvring air targets in strong sea-clutter
and is also applied to both simulated data and real world experimental data. Results
demonstrate the effectiveness of the proposed method in extracting m-D radar signatures of
rotating targets. Its potential as a tool for detecting, enhancing low observable manoeuvring
and accelerating air targets in littoral environments is demonstrated.
Le présent rapport décrit l’efficacité de la méthode fondée sur l’analyse temps-fréquence et
la transformée de Fourier-Bessel, de concert avec la transformée de Fourier fractionnaire,
pour extraire les signatures radar obtenues par microdécalage Doppler dans les cibles
rotatives. Cette approche comprend principalement deux processus, le pre-mier étant la
décomposition des échos radar pour extraire les caractéristiques du microdécalage
Doppler, et la seconde étant l’analyse temps-fréquence pour évaluer les paramètres de
déplacement des cibles. Afin d’extraire les caractéristiques du mi-crodécalage Doppler
dans les échos radar, les signaux radar du domaine temporel sont divisés en éléments fixes
et non fixes à l’aide de la transformée de Fourier-Bessel, de concert avec la transformée de
Fourier fractionnaire. Les éléments sont alors reconstitués en utilisant la transformée
inverse de Fourier-Bessel. Une fois les caractéristiques extraites de l’écho radar original
des cibles, l’analyse temps-fréquence permet d’évaluer les paramètres de déplacement des
cibles. Cette méthode proposée constitue également un outil efficace de détection des
cibles aériennes manIJuvrables dans un important fouillis de mer. Il est également utilisé
pour les données simulées et les données expérimentales réalistes. Les résultats démontrent
l’efficacité de cette méthode pour extraire les signatures radar obtenues par microdécalage
Doppler des cibles rotatives. Son potentiel comme outil de détection et de poursuite de
cibles aériennes manIJuvrables et furtives dans des milieux littoraux est démontré.
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KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could
be helpful in cataloguing the document. They should be selected so that no security classif cation is required. Identif ers, such as
equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords
should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientif c Terms (TEST) and that thesaurus identif ed.
If it is not possible to select indexing terms which are Unclassif ed, the classif cation of each should be indicated as with the title.)
Micro-Doppler; Fourier Bessel transform; Fractional Fourier Transform; Time-Frequency Analysis; High-Frequency Surface-Wave Radar; SAR